Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, q\neq 0$. $\dfrac{{(a^{4})^{-1}}}{{(a^{-1}q^{5})^{-5}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{4}}$ to the exponent ${-1}$ . Now ${4 \times -1 = -4}$ , so ${(a^{4})^{-1} = a^{-4}}$ In the denominator, we can use the distributive property of exponents. ${(a^{-1}q^{5})^{-5} = (a^{-1})^{-5}(q^{5})^{-5}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{4})^{-1}}}{{(a^{-1}q^{5})^{-5}}} = \dfrac{{a^{-4}}}{{a^{5}q^{-25}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-4}}}{{a^{5}q^{-25}}} = \dfrac{{a^{-4}}}{{a^{5}}} \cdot \dfrac{{1}}{{q^{-25}}} = a^{{-4} - {5}} \cdot q^{- {(-25)}} = a^{-9}q^{25}$.